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A fundamental scheme is utilized to efficiently implement the full-vectorial alternating-direction implicit beam-propagation method (ADI-BPM). In the conventional ADI-BPM, four coupled Fresnel equations contain eight derivatives in the right-hand sides (RHSs): four for a specific polarization and four for a polarization coupling. Introduction of the fundamental scheme leads to derivative-free forms for the specific polarization in the RHSs, where only the four mixed derivatives for the polarization coupling are calculated. This results in the computation time reduced by 20%. For further simplification, two out of the four mixed derivatives are replaced with auxiliary variables, where only the remaining two mixed derivatives are taken into account in the RHSs. This reduces the computation time by 34%, with the equivalence to the conventional method.